CP -violation in B → π + π - and the unitarity triangle

. We analyze the extraction of weak phases from CP -violation in B → π + π - decays. We propose to determine the unitarity triangle (ρ̅,η̅) by combining the information on mixing-induced CP -violation in B → π + π - , S , with the precision observable sin2β obtained from the CP -asymmetry in B → ψ K S . It is then possible to write down exact analytical expressions for ρ̅ and η̅ as simple functions of the observables S and sin2β and of the penguin parameters r and ϕ. As an application clean lower bounds on η̅ and 1-ρ̅ can be derived as functions of S and sin2β, essentially without hadronic uncertainty. Computing r and ϕ within QCD factorization yields precise determinations of ρ̅ and η̅ since the dependence on r and ϕ is rather weak. It is emphasized that the sensitivity to the phase ϕ enters only at second order and is extremely small for moderate values of this phase, predicted in the heavy-quark limit. Transparent analytical formulas are further given and discussed for the parameter C of direct CP -violation in B → π + π - . Predictions and uncertainties for r and ϕ in QCD factorization are examined in detail. It is pointed out that a simultaneous expansion in 1/ m b and 1/ N leads to interesting simplifications. At first order infrared divergences are absent, while the most important effects are retained. Independent experimental tests of the factorization framework are briefly discussed.